Two vectors, and , lie in the xy plane. Their magnitudes are 4.25 and 7.45 units, respectively, and their directions are 312° and 86.0°, respectively, as measured counterclockwise from the positive x axis. What are the values of the following products?
A) (r)(s)= -18.3
B) (r)X(s)=?

A) r•s = -21.99 units

B) rXs = 22.77 units (In the direction of k)

Completed question

Two vectors, r and s, lie in the xy plane. Their magnitudes are 4.25 and 7.45 units, respectively, and their directions are 312° and 86.0°, respectively, as measured counterclockwise from the positive x axis. What are the values of the following products?

A) r•s =

B) (r)X(s)=

Explanation:

A) dot product of r and s can be expressed mathematically as;

r•s = |r||s|cos(A) 1

Where A is the angle between the two vectors.

|r| and |s| are the magnitude of vector r and s.

r•s = 4.25×7.45 ×cos(86-312) = -21.99 units

B) cross product of r and s can be expressed mathematically as;

rXs = |r||s|sin(A) 2

Where A is the smallest angle between the two vectors

|r| and |s| are the magnitude of vector r and s.

A = 312-86 = 226

A = 360-226 = 134 smallest

rXs = 4.25 × 7.45 × sin134.

rXs = 22.77 units

In the direction of k

(a) -22.00 units

(b) 22.77 units

Given vectors are r and s

Where;

r = |r| = 4.25   and ∠r = 312°    measured anticlockwise

s = |s| = 7.45   and ∠s = 86°    measured anticlockwise

First, let's calculate the angle between vectors r and s by representing them in the figure below;

                                y |        /s

                                   |      /

                                   |    /

                                   |  /  

                                   |/ )86°                                x

                                   |\) 48°

                                   |  \

                                   |    \

                                   |      \

                                   |        \ r

To get the acute angle between r and the +x axis, subtract the reflex angle of r (312°) from 360° as follows;

360 - 312 = 48°

As shown in the diagram, the angle between vectors r and s is 48° + 86° = 134°

Now,

(a) The (r)(s) represents the dot or scalar product of the two vectors and it is given as;

(r) (s) = r x s cos θ            (i)

Where;

r = magnitude of vector r = 4.25

s = magnitude of vector s = 7.45

θ is the angle between the two vectors r and s = 134°

Substitute these values into equation (i) as follows;

(r) (s) = 4.25 x 7.45 cos 134°

(r) (s) = 4.25 x 7.45 x -0.6947

(r) (s) = 31.66 x -0.6947

(r) (s) = -22.00 units

(b) The (r) X (s) represents the vector product of the two vectors and it is given as;

(r) (s) = r x s sin θ            (ii)

Where;

r = magnitude of vector r = 4.25

s = magnitude of vector s = 7.45

θ is the angle between the two vectors r and s = 134°

Substitute these values into equation (ii) as follows;

(r) (s) = 4.25 x 7.45 sin 134°

(r) (s) = 4.25 x 7.45 x 0.7193

(r) (s) = 31.66 x 0.7193

(r) (s) = 22.77 units

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